I have a circle where all the points inside a circle are labelled positively and all the points outside the circle are labelled negatively. The center of the circle is h and radius is r. Let H+ be the class of the positive circle.
I need to prove vc dimension of $H+\geq 3$. Any suggestions.
From the definition of vc dimension I know that $mH(N)=2^N$
Consider the points $(1/4,0),$ $(-1/4,0),$ and $(0,2).$ It is is obvious that we can find circles that classify all three points positive (some big circle containing all three), all three negative (some tiny circle far away from the origin), and each individually positive (just take a tiny circle around that point), so the only subsets we need to worry about are the pairs.
It's clear we can find a circle containing the first two but not the third (say the circle centered on the origin with radius $1/2$).
For classifying only $(1/4,0)$ and $(0,2)$ positive, take a circle with a center far from the origin in the first quadrant, that looks pretty much like a straight line sloping down and to the right as it passes in between $(1/4,0)$ and $(-1/4,0)$ while still containing $(0,2)$ (it shouldn't be too hard to get an explicit equation for a circle that works here). For the remaining pair, take the mirror image of the previous circle.
(In fact, one can generalize this last example to show something stronger: that any three non-colinear points can be shattered by the circle classifier.)